Here’s a quiz
A healthy woman with no risk factors gets a positive mammogram result during a routine annual check. What is the probability that she actually has a breast cancer?
Baseline data: The probability that a woman has breast cancer is 0.8%. If she has breast cancer, the probability that a mammogram will show a positive result is 90%. If a woman does not have breast cancer, the probability of a positive result is 7%.
Prof. Gerd Gigerenzer gave this quiz to numerous students, physicians, and professors. Most of them failed this quiz. The correct answer is 9%. The probability that a healthy woman has a breast cancer if she has a positive mammogram test is only nine percent! This means that ninety percent of women who get a positive result will undergo stressful and painful series of tests only to discover that that was a false alarm. In his book “Calculated Risks“, prof. Gigerenzer uses this low probability as a starting (but not the only) argument against the common practice of routine population-wide mammogram tests. However, I would like to propose another way to look at this problem.
To understand my concern, let me first explain how we get the 9% figure.
There are several ways to get to this result. One of them is as follows. Eighty out of 10,000 women have breast cancer. Of those women, 72 (90% of 80) will test positive during a mammogram. Of the remaining 9,920 healthy women, about 694 (7%) will also have a positive mammogram test. The total number of women with a positive test is 766. Of those 766 women, only 72 have breast cancer, which is about 9%. The following diagram will help you track the numbers.
Nine percent is indeed a low number. If a woman gets ten mammogram tests in her lifetime, there is a 60+% chance that she will have at least one false positive test. This is not something that can be easily ignored.
Let’s think about another way to look at this problem. Yes, the probability of a woman to have a breast cancer given that she has a positive mammogram result is nine percent (72 out of 697+72=766). However, the probability of a woman to have a breast cancer given that she has a negative mammogram result is 8 out of (9,223+8)=9,231 which is approximately 0.09%. That means that a woman with a positive mammogram test is 100 times more likely to have a breast cancer, compared to the woman with a negative result. Increase by a factor of 100 sounds like a serious threat. Much more serious than the nine percent! Moreover, a woman with a negative mammogram result knows that she is approximately ten times less likely to have a breast cancer than an average woman who didn’t undergo the test (0.09% vs 0.8%).
Frankly, I don’t know. One thing is for sure; one can use statistics to steer an “average person” towards the desired decision. If my goal is to increase reduce the number of women who undergo routine mammogram tests, I will talk in terms of absolute risk (9%). If, on the other hand, I’m selling mammogram equipment, I will definitely talk in terms of the odds ratio, i.e., the 100-times risk increase. Think about this every time someone is talking to you about hazards.